Limit of a Function Philippe B. Laval

Transcription

1 Limit of a Function Philippe B. Laval

2 Limit of a Function 2 1 Limit of a Function 1.1 Definitions and Elementary Theorems Unlike for sequences, there are many possibilities for the limit of a function. In this section, we will investigate the following limits: lim f (x) and lim f (x). x ± Limit at a finite point In order to be able to evaluate lim f (x), f must be definedina deleted neighborhood of a that is f must be definedinaninterval of the form (a h, a + h) for some positive number h,except maybe at x = a. We have the following definitions: Definition 1 We say that lim f (x) =L or that f (x) L as x a if ɛ >0, δ >0:0< x a <δ= f (x) L <ɛ. x a represents how far x is from a. The above statement says that f (x) can be made arbitrarily close to L simply by taking x close enough to a. Example 1 Prove that lim x 2 (x +5)=7. Remark 1 Of course, this was a very easy example to illustrate how this kind of problem is addressed. In general, it will take more work to find δ given ɛ>0. Many of the techniques used for

3 Limit of a Function 3 sequences will also be used here. Some will be illustrated below when we look at more challenging examples. Definition 2 We say that lim f (x) = or that f (x) as x a if M >0, δ >0:0< x a <δ= f (x) >M. Example 2 Prove that lim x 1 1 (x 1) 2 =. Definition 3 We say that lim f (x) = or that f (x) as x a if M <0, δ >0:0< x a <δ= f (x) <M. Remark 2 If we find a δ which works, then every δ <δwill also work. Therefore, it is always possible to impose certain conditions on δ such as saying that we are looking for δ less than a certain number h, thus restricting our search to an interval of the form (a h, a + h). In this interval, if we call δ the value we found, then δ =min ( h, δ ). Remark 3 In the last two definition, the vertical line x = a is a vertical asymptote for the graph of y = f (x). Remark 4 In the definition of a limit, it is implied that a is a limit point of D (f) that is for every δ>0 the interval (a δ, a + δ)

4 Limit of a Function 4 contains points of D (f) other than a. Ifthisisnotthecase, then for δ small enough, there may not exist any x satisfying 0 < x a < δ. In this case, the concept of a limit has no meaning. Remark 5 In the definition of a limit, a does not have to be in the domain of f. It only needs to be a limit point of D (f). Remark 6 In the definition of a limit, δ depends obviously on ɛ. It may also depend on the point a as illustrated by example 10. Remark 7 When we say that the limit of a function exists, we mean that it exists and is finite. When the limit is infinite, it does not exist in the sense that it is not a number. However, we know what the function is doing, it is approaching ±. Remark 8 There are several situations under which a limit will fail to exist. (1) The function may oscillate boundedly like in f (x) =sin 1 x as x 0. (2) The function may oscillate unboundedly like x sin x as x.

5 Limit of a Function 5 (3) The function may grow without bounds like 1 as x 0. x Limit at infinity In order to be able to evaluate lim f (x), f must be defined for x large x. Inotherwords,wemusthaveD (f) (w, ) for every w R. In the case we want to evaluate lim x f (x), then we must have D (f) (,w) for every w R. Wethen have the following definitions (some of the definitions will be accompanied with easy examples to illustrate the concept being defined): Definition 4 We say that lim x f (x) = L or that f (x) L as x if ɛ >0, w > 0:x (w, ) D (f) = f (x) L <ɛ f (x) L represents the distance between f (x) and L. The above statement simply says that f (x) can be made as close as one wants from L, simply by taking x large enough. Graphically, this simply says that the line y = L is a horizontal asymptote for the graph of y = f (x). To prove that a number f (x) approaches L as x, given ɛ>0, one has to prove that w>0can be found so that x (w, ) D (f) = f (x) L <ɛ. The approach is very similar to the one used for sequences. Many of the techniques used when finding the limit of a sequence will also be used here.

6 Limit of a Function 6 1 Example 3 Prove that lim x x =0. Definition 5 We say that lim x f (x) = or that f (x) as x if M >0, w >0:x (w, ) D (f) = f (x) > M. The above definition says that f (x) can be made arbitrarily large, simply by taking x large enough. Example 4 Prove that lim x x 2 =. Definition 6 We say that lim x f (x) = or that f (x) as x if M < 0, w >0:x (w, ) D (f) = f (x) <M. Definition 7 We say that lim f (x) =L or that f (x) L as x x if ɛ >0, w <0:x (,w) D (f) = f (x) L <ɛ. Definition 8 We say that lim f (x) = or that f (x) x as x if M >0, w <0:x (,w) D (f) = f (x) >M. Definition 9 We say that lim f (x) = or that f (x) x as x if M < 0, w < 0 : x (,w) D (f) = f (x) <M.

7 Limit of a Function One-sided Limits When we say x a, we realize that x can approach a from two sides. If x approaches a from the right, that if x approaches a and is greater than a, we write x a +. Similarly, if x approaches a from the left, that is if x approaches a andislessthana, thenwe write x a. We can rewrite the above definition for one sided limits with little modifications. We do it for a few of them. Definition 10 We say that lim f (x) =L or that f (x) L as + x a + if ɛ >0, δ >0:0<x a<δ= f (x) L <ɛ Definition 11 We say that lim f (x) =L or that f (x) L as x a if ɛ >0, δ >0:0<a x<δ= f (x) L <ɛ Example 5 Prove that lim x x =. Theorem 1 The following two conditions are equivalent (1) lim f (x) =L (2) lim f (x) =L and lim f (x) =L + Proof. See problems.

8 Limit of a Function 8 Remark 9 One way to prove that lim f (x) does not exits is to prove that the two one-sided limits are not the same or that at least one of them does not exist. Remark 10 One sided limits are often used when the definition or behavior of f changes around the point a at which the limit is being computed. This can happen with piecewise functions when we compute their limit at one of the breaking points Examples Example 6 Show that lim x 3 (4x 5) = 7 Example 7 Show that lim x 2 ( x 2 +2 ) =6 Remark 11 In the previous example, the choice of the interval (0, 4) is not magic. We could have chosen another interval. We want to use an interval centered at the point where we are computing the limit. Some readers may think we cheated by only looking at values of x in some intervals. Remember what we are trying to achieve. Given an ɛ>0, we are finding δ>0 with certain properties. As long as we find a δ,wehaveachievedwhatwehad to achieve. By picking an interval, we simply acknowledge the

9 Limit of a Function 9 fact that it is too difficult to look for just any δ, so we restrict our search to a smaller interval. Our next example illustrates how to work with piecewise functions. Example 8 Let f (x) = x if x<0 x 2 if 0 <x 2 8 x if x>2 (1) Prove that lim x 0 f (x) =0. (2) Prove that lim x 2 f (x) doe not exist. Example 9 Find lim x 4 x 2 x 4. The next example illustrates the fact that δ depends not only on ɛ but also on the point at which the limit is being found. 1 Example 10 Prove that lim x = 1 a for any a (0, ). Example 11 Let f : R R defined by { 1 if x Q f (x) = 0 if x/ Q Prove that lim f (x) does not exist for any a R.

10 Limit of a Function Elementary Theorems Theorems similar to those studied for sequences hold. We will leave the proof of most of these as an exercise. Theorem 2 If the limit of a function exists, then it is unique. The next theorem relates the notion of limit of a function with the notion of limit of a sequence. Theorem 3 Suppose that lim f (x) =L and that {x n } is a sequence of points such that lim x n = a, x n a n. Put y n = n f (x n ).Then, lim y n = L n The converse of this theorem is also true. Theorem 4 If f is defined in a deleted neighborhood of a such that f (x n ) L for every sequence {x n } such that x n a,then f (x) =L. lim Like for sequences, if a function has a limit at a point, then it is bounded. However, we need to be a little bit more careful here. If lim f (x) =L, then we are only given information about the behavior of f close to a. Therefore, we can only draw conclusions about what happens to f as long as x is close to a.

11 Limit of a Function 11 Theorem 5 If lim f (x) =L, then there exists a deleted neighborhood of a in which f is bounded. Theorem 6 Suppose that f (x) g (x) h (x) in a deleted neighborhood of a and lim f (x) =limh (x) =L then lim g (x) = L. 1.2 Operations with Limits The same theorem we proved for sequences also hold for functions. We list the theorem, and leave its proof as an exercise. Theorem 7 Assuming that lim f (x) and lim g (x) exist, the following results are true: (1) lim (f (x) ± g (x)) = lim f (x) ± lim g (x) ( )( ) (2) lim (f (x) g (x)) = lim f (x) lim g (x) f (x) lim (3) lim g (x) = lim (4) lim f (x) = f (x) as long as lim g (x) g (x) 0 lim f (x) (5) If f (x) 0,thenlim f (x) 0 (6) If f (x) g (x) then lim f (x) lim g (x)

12 Limit of a Function 12 (7) If f (x) 0,thenlim f (x) = lim f (x) 1.3 Relationship Between the Limit of a Function and the Limit of a Sequence Theorems 3 and 4 can be summarized in the theorem below. Theorem 8 Let f be a function of one real variable definedina deleted neighborhood of a real number a. The following conditions are equivalent. (1) lim f (x) =L (2) For every sequence {x n } such that x n a and x n a we have lim n f (x n )=L. 1.4 Exercises (1) Use the definition of the limit of a function to show that x = 2 lim x 2 (2) Use the definition of the limit of a function to show that 1 lim x 2 x = 1 2 (3) Prove theorem 2 (4) Prove theorem 7 (5) Evaluate the following limits:

13 Limit of a Function 13 (a) lim x 2 x 2 4 x +2 (b) lim x 3 x 3 27 x 3 x n 1 (c) lim x 1 x 1 x n 1 (d) lim x 1 (e) lim x 6 x m 1 x 2 2 x 6 where n is a positive integer. sin x (6) Assuming you know that lim x 0 x below: sin 2x (a) lim x 0 x sin 3x (b) lim x 0 sin 5x x (c) lim x 0 tan x sin x (d) lim x 0 x (7) Evaluate lim x 0 (a + bx)(c + dx) ac x =1, compute the limits (8) Discuss the one-sided limits of f (x) =e 1 x at x =0. (9) Write a careful proof of the results stated below.

14 Limit of a Function 14 ( (a) lim x 3 3x ) =2 x 1 1 (b) lim x 3 x = 1 3 x 3 8 (c) lim x 2 x 2 + x 6 = 12 5 { x if if x is rational (10)Let f (x) = x 2 Prove the following: if if x is irrational (a) lim f (x) =1 x 1 (b) lim f (x) does not exist. x 2 (11)Prove theorem 1. (12)Prove Theorem 4. (13)We say that a function f : I R is Lipschitz providing there exists a constant K>0 such that f (x) f (y) K x y for x, y I. (a) Give an example of a Lipschitz function. You must show that the function in your example satisfies the required condition. (b) Prove rigorously that if f is Lipschitz, then lim f (x) = f (a).